Method for reconstructing non-uniform circumferential flow in gas turbine engines

ABSTRACT

A method for reconstructing nonuniform circumferential flow in a turbomachine is disclosed which includes receiving one or more wavenumbers of interest, receiving positional information for a plurality of circumferential positions of a plurality of instrumentation probes, receiving signals from the plurality of instrumentation probes to generate a spatially under-sampled data, and from the spatially under-sampled data determining a multi-wavelet approximation reconstructing circumferential flow field.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present patent application is related to and claims the priority benefit of U.S. Provisional Patent Application Ser. No. 63/073,029, entitled METHOD FOR RECONSTRUCTING NON-UNIFORM CIRCUMFERENTIAL FLOW IN GAS TURBINE ENGINES, filed Sep. 1, 2020, and U.S. Provisional Patent Application Ser. No. 63/073,024, entitled PROBE PLACEMENT OPTIMIZATION IN GAS TURBINE ENGINES, filed Sep. 1, 2020, the contents of which are hereby incorporated by reference in its entirety into the present disclosure.

STATEMENT REGARDING GOVERNMENT FUNDING

None.

TECHNICAL FIELD

The present disclosure generally relates to gas turbine engines and in particular, to an optimized methodology of probe placement to measure the mean flow properties such as temperature and pressure.

BACKGROUND

This section introduces aspects that may help facilitate a better understanding of the disclosure. Accordingly, these statements are to be read in this light and are not to be understood as admissions about what is or is not prior art.

A flow field in a compressor is circumferentially non-uniform. The circumferential variations measured in an absolute reference frame are associated with the wakes from upstream stator row(s), potential fields from both upstream and downstream stator rows, and their aerodynamic interactions. In a typical engine or technology development programs, the performance such as thermal efficiency of the engine or component is commonly characterized using measurements acquired from a few probes at different circumferential locations. However, because the flow in a gas turbine engine is non-uniform along the circumferential direction, the calculated engine performance using measurements from one probe set can be different from another probe set with changes in the circumferential locations.

Also, stator-stator and rotor-rotor interactions can impact stage performance. For example, in a 2.5-stage transonic axial compressor a 0.1% efficiency variation was seen due to stator-stator interactions and a maximum of 0.7% variation in efficiency was observed caused by rotor-rotor interactions. The effect of stator-stator interactions on stage performance have been investigated using vane clocking, the circumferential indexing of adjacent vane rows with the same vane count. According to another example, in a 3-stage axial compressor a 0.27-point variation in the isentropic efficiency of the embedded stage was observed at the design loading condition and a 1.07-point variation in the embedded stage efficiency was observed at a high loading condition with changes in vane clocking configurations. The experimental characterization of stage efficiency is facilitated when similar vane counts exist because that means that measuring the flow across a single vane passage will accurately capture the full-annulus performance. This is great for research, but it is not a common luxury for real compressors, in which the stators typically have different vane counts requiring measurements over several pitches, if not the entire annulus, to accurately capture the circumferential flow variations.

Once probe placements have been identified, what is then needed is a method that provides mean flow properties for performance calculations as well as resolving the important flow features associated with circumferential non-uniformity to thereby calculate suitable mean flow properties of the turbomachine.

Therefore, there is an unmet need for a novel approach for a method that can provide accurate mean flow properties for performance calculations in a turbomachine.

SUMMARY

A method for reconstructing nonuniform circumferential flow in a turbomachine is disclosed. The method includes receiving one or more wavenumbers of interest, receiving positional information for a plurality of circumferential positions of a plurality of instrumentation probes, receiving signals from the plurality of instrumentation probes to generate a spatially under-sampled data, and from the spatially under-sampled data determining a multi-wavelet approximation reconstructing circumferential flow field.

BRIEF DESCRIPTION OF DRAWING

FIG. 1 is a schematic of a typical gas turbine engine.

FIG. 2 is a graph of circumferential total pressure field at mid-span upstream of stator number 6 in a multi-stage axial compressor at different angular positions with probes shown.

FIG. 3 is a schematic which shows the circumferential total pressure field at mid-span upstream of stator number 6 in a multi-stage axial compressor.

FIGS. 4 and 5 show the values of Pearson's correlation coefficient r, also provided in the present disclosure as p, (FIG. 4) and the root-mean-square of the fitting residual (FIG. 5) for all wave number combinations.

FIG. 6a is similar to FIG. 2 and shows both the data from the true measurements and the reconstruction using a multi-wavelet approximation.

FIG. 6b is related to FIG. 6a and shows the deviation between the two data sets, showing there is significant improvement in data fitting with the additional wavelets.

FIGS. 7a and 7b show comparison of the reconstructed total pressure field from the best cases of single-, double-, and triple-approximation with the true total pressure field in the spatial domain (FIG. 7a ) and the comparison in magnitudes at specific wavenumbers (FIG. 7b ).

FIG. 8a is a schematic of a flow path of a compressor and distribution of the steady instrumentation.

FIG. 8b is a diagram depicting distribution of the static pressure taps at a diffuser leading edge.

FIG. 9 is a graph of normalized total pressure ratio (TPR) vs. normalized corrected mass flow rate which shows the performance map of the compressor stage in terms of normalized TPR from 60% to 100% corrected speed from choke to high loading conditions.

FIGS. 10a and 10b show the variation of the diffuser leading edge static pressure, wherein in FIG. 10a , the static pressure is shown in terms of the absolute circumferential locations of the measurements and in FIG. 10b the static pressure data is shown in terms of its pitchwise position.

FIG. 11 is a graph of condition number vs. wavenumber for the 10 selected wavenumbers.

FIG. 12 is a graph similar to FIG. 2, where the deviation between the measurements and fitting results at individual probe locations are provided.

FIG. 13a is a graph of the reconstructed pressure field at the leading edge from the triple-wavelet approximation method.

FIG. 13b is a graph of the reconstructed pressure field at the leading edge of the diffuser determined by constructive and destructive interactions between three wavelets.

FIG. 13c is a graph of the reconstructed pressure field showing pressure variation due to the presence of diffuser vanes (Wn=25) is approximately 14% with respect to the pitch-averaged value.

FIGS. 14a and 14b are comparisons of errors in calculating mean diffuser leading edge (FIG. 14a ) and static pressure and static pressure recovery coefficient (FIG. 14b ) using averaging methods and the multi-wavelet approximation method.

FIG. 15 is a schematic of a second actual reduction to practice using a PAX100 compressor with a reduced vane count for stator 1 with an inlet guide vane (IGV) followed by three stages.

FIG. 16 is a schematic of a simple model to demonstrate how the blades would line up relative to one another around an annulus.

FIG. 17 is a graph of offset angles of Si and S3 with respect to S2/IGV at all 7 clocking configurations.

FIG. 18 is a graph of normalized total pressure vs. normalized inlet corrected mass flow at 86% corrected speed on the 100% corrected speed peak efficiency loading line.

FIG. 19 is a graph of the total pressure measurements acquired at mid-span downstream of S2.

FIG. 20a is a reconstructed total pressure profile at mid-span downstream of S2, with the total pressure profile acquired from experiments plotted on top of the reconstructed total pressure profile for comparison.

FIG. 20b is a graph of total pressure at mid-span downstream of S2 that was reconstructed using the first 12 dominant wavelets.

FIG. 20c is a graph of the reconstructed total pressure field using a reduced dataset.

FIGS. 21a, 21b, and 21c are the total pressure profile at near hub (12% and 20% shown in FIG. 21a ), 35% and 50% span shown in FIG. 21b , and near shroud (80% and 88%, shown in FIG. 21c ) which are all reconstructed using the reduced dataset with 12 wavelets.

FIG. 22 is a flowchart of the method of the present disclosure.

FIG. 23 is a schematic of a typical multi-stage axial compressor used in high-pressure compressor (HPC) assembly of gas turbines.

DETAILED DESCRIPTION

For the purposes of promoting an understanding of the principles of the present disclosure, reference will now be made to the embodiments illustrated in the drawings, and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of this disclosure is thereby intended.

In the present disclosure, the term “about” can allow for a degree of variability in a value or range, for example, within 10%, within 5%, or within 1% of a stated value or of a stated limit of a range.

In the present disclosure, the term “substantially” can allow for a degree of variability in a value or range, for example, within 90%, within 95%, or within 99% of a stated value or of a stated limit of a range.

A novel approach for a method that can provide accurate mean flow properties for performance calculations in a turbomachine is provided. Towards this end, the present disclosure provides a multi-wavelet approximation method to reconstruct the non-uniform circumferential flow from several dominant wavenumbers.

A gas turbine engine typically includes three elements including: a compressor, a combustor, and a turbine. Referring to FIG. 1, a schematic of a typical gas turbine engine 100 is shown. The gas turbine engine 100 typically includes an intake 102 which includes an air intake 104 having an initial cross sectional area through which air is allowed into the gas turbine engine 100 at a high rate of speed. The incoming air is compressed through a compressor 106 which reduces the effective cross section prior to entering into a combustion zone 108 having one or more combustion chambers 110. In the one or more combustion chambers 110, the compressed air is energized and then is directed to turbines 112 prior to being ejected out of an exhaust 114.

The compressor 106 or turbines 112 include stationary blade rows which are called stators as well as rotating blade rows which are called rotors. Each includes a plurality of stages. Thus, a stage includes a stator and a rotor. The flow field in a compressor or turbine is circumferentially non-uniform due to the wakes from upstream stators, the potential field from both upstream and downstream stators, and blade row interactions. To demonstrate this non-uniformity, reference is made to FIG. 3 which shows the circumferential total pressure field at mid-span upstream of stator number 6 in a multi-stage axial compressor. The abscissa represents the circumferential position along the full annulus and the ordinate represents the normalized nondimensional pressure. As described above, this observed non-uniformity is due to the wakes from upstream stator row(s), potential fields from both upstream and downstream stator rows, and their aerodynamic interactions.

In theory, the circumferential flow field in turbomachines with a spatial periodicity of 2π can be described in terms of infinite serial wavelets of different wavenumbers:

x(θ)=c ₀+Σ_(i=1) ^(∞)(A _(i) sin(W _(n,i)θ+φ_(i)))   (1)

-   where x(θ) represents the flow property along the circumferential     direction, -   c₀ represents the DC component of the signal, -   W_(n,i) represents the i^(th) wavenumber, and -   A_(i) and φ_(i) represent the magnitude and phase of the wavelet of     the i^(th) wavenumber.

Furthermore, defining a_(i)=A_(i) cos φ_(i) and b_(i)=A_(i) sin φ_(i), Eqn. (1) can be cast as:

x(θ)=c ₀+Σ_(i=1) ^(∞)(a _(i) sin(W _(n,i)θ)+b _(i) cos(W _(n,i)θ)).   (2)

The circumferential flow in a multi-stage compressor is typically dominated by several wavenumbers. Therefore, instead of using an infinite number of wavelets described in Eqn. (1), the circumferential flow in the compressor can be approximated by a few (N) dominant wavelets (where the dominance is measured by the magnitude based on a predetermined threshold weight of magnitude):

x(θ)≈c ₀+Σ_(j=1) ^(N)(a _(j) sin(W _(n,j)θ)+b _(j) cos(W _(n,j)θ)).   (3)

Furthermore, Eqn. (3) can be described:

AF=x,   (4)

-   where A is known as the design matrix with a dimension of m×(2N+1),     which are functions of wavenumber of interest, W_(n), and probe     displacement, θ, -   F is a vector containing 2N+1 unknown coefficients, and -   x is an m-element vector with all the measurement data points (i.e.,     measurements from probes) from different circumferential locations.     The mathematical expressions for A, F, and x are

${A = \begin{pmatrix} {\sin\mspace{14mu} W_{n,1}\theta_{1}} & {\cos\mspace{14mu} W_{n,1}\theta_{1}} & \cdots & {\sin\mspace{14mu} W_{n,N}\theta_{1}} & {\cos\mspace{14mu} W_{n,N}\theta_{1}} & 1 \\ {\sin\mspace{14mu} W_{n,1}\theta_{2}} & {\cos\mspace{14mu} W_{n,1}\theta_{2}} & \cdots & {\sin\mspace{14mu} W_{n,N}\theta_{2}} & {\cos\mspace{14mu} W_{n,N}\theta_{2}} & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {\sin\mspace{14mu} W_{n,1}\theta_{m}} & {\cos\mspace{14mu} W_{n,1}\theta_{m}} & \cdots & {\sin\mspace{14mu} W_{n,N}\theta_{m}} & {\cos\mspace{14mu} W_{n,N}\theta_{m}} & 1 \end{pmatrix}};$ $\mspace{76mu}{{F = \begin{pmatrix} a_{1} \\ b_{1} \\ \vdots \\ a_{N} \\ b_{N} \\ c_{0} \end{pmatrix}};}$ $\mspace{76mu}{x = \begin{pmatrix} {x\left( \theta_{1} \right)} \\ {x\left( \theta_{2} \right)} \\ \vdots \\ {x\left( \theta_{m} \right)} \end{pmatrix}}$

To solve for the N wavenumbers of interest described in Eqn. (4), the number of the data points in vector x must be equal or greater than the number of unknown coefficients, or m≥2N+1. However, in practice, the reconstructed signal contains errors due to the uncertainties in x(θ), and it is important to evaluate the confidence in the reconstructed signal, which requires additional data points in x(θ). Therefore, a minimum of 2N+2 measurement points is required to characterize N wavenumbers of interest. The unknown coefficients, F, is calculated:

F=A\x ,   (5)

where “\” represents inverse matrix operation.

The circumferential flow field can be reconstructed using Eqn. (3) and the mean value of the flow is:

f_(mean)=c₀.   (6)

An illustration of reconstruction of the circumferential flow field from spatially under-sampled data using the multi-wavelet approximation method is shown in FIG. 3, which is a schematic showing a reconstructed circumferential flow approximation based on placement of a plurality probes (sensors) along the turbomachine.

It is important to gauge the confidence in the reconstructed circumferential flow obtained from multi-wavelet approximation method. To achieve this objective, two parameters, including the Pearson correlation coefficient and root-mean-square of the fitting residual, are utilized. The Pearson correlation coefficient, or Pearson's r, is a measure of the linear correlation between two variables. Its magnitude varies between 0 and 1, with values close to 1 indicating a strong linear correlation. In the present method, the two variables used for correlation are the predicted flow properties, x_(fit)(θ), from the reconstructed signal and the true measurements, x(θ). The formula for calculating the Pearson correlation coefficient is:

$\begin{matrix} {\rho = {\frac{{\sum\limits_{j = 1}^{m}\;{x_{j}x_{{fit},j}}} - {\left( {\sum\limits_{j = 1}^{m}\;{x_{j}{\sum\limits_{j = 1}^{m}\; x_{{fit},j}}}} \right)\text{/}m}}{\sqrt{\left( {{\sum\limits_{j = 1}^{m}\; x_{j}^{2}} - {\left( {\sum\limits_{j = 1}^{m}\; x_{j}} \right)^{2}\text{/}m}} \right)}\left( {{\sum\limits_{j = 1}^{m}\; x_{{fit},j}^{2}} - {\left( {\sum\limits_{j = 1}^{m}\; x_{{fit},j}} \right)^{2}\text{/}m}} \right)}.}} & (7) \end{matrix}$

-   where ρ is the Pearson correlation coefficient, -   m represents the number of measurements, -   x_(j) is the measurement at the j^(th) circumferential location, and -   x_(fit,j) corresponds to the reconstructed value at the j^(th)     circumferential location.

For a well-reconstructed circumferential flow field, the predicted flow properties will align with true values at all the measurement locations and yield a value of nearly 1 for the Pearson's r. In contrast, the predicted flow properties from a poorly reconstructed flow field will deviate from the true measurements resulting in a small value for Pearson's r.

In addition to Pearson's r, the confidence in the reconstructed flow can also be evaluated in terms of the root-mean-square of the fitting residual between the reconstructed signal and true measurements. The formula for calculating R_(rms) is:

$\begin{matrix} {R_{rms} = {\sqrt{\frac{1}{m}\left( {\sum\limits_{j = 1}^{m}\;\left( {x_{{fit},j} - x_{j}} \right)^{2}} \right)}.}} & (8) \end{matrix}$

To demonstrate the efficacy of this novel approach, an actual reduction to practice was carried out with probe positioned as shown in FIG. 2 with circle representing the position of probes. One evident feature associated with the optimized probe set is that they are not equally spaced. The maximum probe spacing falls between P3 and P4, with a value of 62°, while the minimum probe spacing is 20°. This non-uniform probe spacing allows for characterization of all wavenumbers of interest.

FIGS. 4 and 5 show the values of Pearson's r (FIG. 4) and the root-mean-square of the fitting residual (FIG. 5) for all wave number combinations. The predicted optimal wavenumber combinations for single-, double-, and triple-wavelet approximations are shown with dotted boxes. For the single-wavelet approximation, the predicted dominant wave number, in terms of highest Pearson's r and smallest fitting residual, equates to the vane count of Stator 5 (S5, vane count 96). For the double-wavelet approximation, the predicted dominant wavenumbers are S5 and S6-S5. Comparing to the single-wavelet approximation using wavenumber of S5, inclusion of the second wavenumber S6-S5 yields higher confidence in the reconstructed signal (higher value of Pearson's r and smaller residual error). At last, a wavenumber set of S5, S6-S5, and S6 (vane count 104) yields the highest fitting confidence and smallest fitting residual among all the 7 wavenumber combination.

Table 1 lists the values of Pearson's r and fitting residual, as well as the rank of individual wavenumber combination. The importance of all the wavenumbers of interest can be quantified and correctly ranked using two parameters: the Pearson correlation coefficient and the fitting residual. Finally, the trend in the value of the Pearson correlation and fitting residual from single- to multi-wavelet approximation can gauge the necessity of including additional wavenumbers. For example, in the present case, the fitting confidence in terms of Pearson's correlation is 99.9% with a fitting residual of less than 0.1% after including three wavenumbers and, thus, indicates little need to include additional wavenumbers.

TABLE 1 Rank of data fitting for a variety of wavenumber combinations Wave No. Pearson Fitting Combination Correlation Residual Rank  8 (S6-S5) 0.366 1.482 5  96 (S5) 0.903 0.683 4 104 (S6) 0.251 1.541 7 [8, 96] 0.976 0.347 2 [8, 104] 0.437 1.433 6 [96, 104] 0.926 0.601 3 [8, 96, 104] 0.999 0.070 1

Furthermore, details of true data and fitting data at all measurement locations from the best cases using single-, double-, and triple-wavelet approximations are shown in FIGS. 6a and 6b . FIG. 6a shows both the data from the true measurements and the reconstruction using the multi-wavelet approximation. FIG. 6b shows the deviation between these two data sets. There is significant improvement in data fitting with the additional wavelets. For example, the best case from the single-wavelet approximation still has large deviations between the fitting and true data at almost all measurement locations (except for P2), resulting in a maximum of 4 points in variation between the fitting data and true measurements, as indicated by the bands in FIG. 6b . The data from double-wavelet approximation show significant improvement in matching the true measurements. Furthermore, the fitting using the three-wavelet approximation shows good agreement with the true measurements at all the probe locations. The variation between the fitting data and true measurements is ten times smaller (<0.3 points) compared to the results from single-wavelet approximation.

Comparison of the reconstructed total pressure field from the best cases of single-, double-, and triple-approximation with the true total pressure field are shown in FIGS. 7a and 7 b. FIG. 7a shows the comparison in the spatial domain while FIG. 7b shows the comparison in magnitudes at specific wavenumbers. There is significant difference in the reconstructed signal using single-wavelet approximation from the true signal, and this is due to the absence of low wavenumber (S6-S5) components. The deviation between the reconstructed and true signal is much reduced in the results using the double-wavelet approximation. Finally, the reconstructed circumferential total pressure obtained from a triple-wavelet approximation shows very good agreement with the true total pressure field.

Additionally, the predicted magnitude at specific wavenumbers from the single-, double-, and triple-approximation methods show fairly good agreement with the true signal, as shown in FIG. 7b . The largest error occurs at the low wavenumber (S6-S5) with a 32.5% over-prediction in magnitude. The errors in the predicted magnitudes at the two large wavenumbers, S5 and S6, are within 5%, as shown in Table 2. There is less error in the predicted phase magnitude, with a maximum error of less than 10% at wavenumber S6.

TABLE 2 Comparison of magnitude and phase at wavenumbers of interest from multi-wavelet approximation with true signal Wave Magnitude, % Phase, rad No. True Fitting Error True Fitting Error  8 0.62 0.83 32.5 5.42 5.23 3.6 (S6-S5)  96 2.22 2.12 4.4 4.19 4.13 1.4 (S5) 104 0.49 0.49 0.8 3.96 3.61 8.9 (S6)

In sum, the circumferential total pressure field in a multi-stage compressor representative of small core compressors is reconstructed using a few spatially under-sampled data points. According to one embodiment, 8 probes were selected for reconstruction of the circumferential flow. Following that, the circumferential locations of the 8 probes were carefully selected using the PSO algorithm. The PSO algorithm can optimize probe positions leading to small condition numbers for all the wavenumber of interest. The circumferential total pressure is reconstructed from the 8 data points using a triple-wavelet approximation and very good agreement between the reconstructed signal and the true signal was achieved.

Two experiments were further curried out to show the feasibility of the method of the present disclosure. The first experiment's objective is to reconstruct the pressure field at the diffuser leading edge using measurements from nine static pressure taps. The flow path of the compressor and distribution of the steady instrumentation is shown in FIG. 8a . The entire stage includes an inlet housing, a transonic impeller, a vaned diffuser, a bend, and deswirl vanes. The inlet housing delivers the flow to the impeller eye. The impeller is backswept and has 17 main blades plus 17 splitters. The diffuser consists of 25 aerodynamically profiled vanes. The compressor design speed is about 45,000 rpm, and the entire stage produces a total pressure ratio near 6.5 at design condition. Steady performance of the compressor stage is characterized using the total pressure and total temperature measurements at compressor inlet (station 1) and deswirl exit (station 5), and static pressure taps are located throughout the flow passage to characterize the stage and component static pressure characteristics, as shown in FIG. 8 a.

The diffuser leading edge static pressure measurements are selected as the focus for this study for four reasons:

-   1. The pressure field at the diffuser leading edge is primarily     dominated by the potential field of the diffuser vanes, and thus     provides an ideal case to examine the methodology; -   2. The instrumentation is readily available in the original     experiment setup; -   3. The performance of the vaned diffuser is evaluated in terms of     static pressure recovery coefficient, and precise calculation for     the mean static pressure at the diffuser leading edge is of great     value in evaluating the aerodynamic performance of the vaned     diffuser; -   4. The diffuser potential field is one of the primary forcing     functions for impeller forced response at resonance.

The distribution of the static pressure taps at the diffuser leading edge is shown in FIG. 8b . There are a total of nine static pressure taps placed non-uniformly along the circumferential direction. Each of them is placed in a different diffuser passage at a different pitchwise location from 10% to 90% pitch. Details of the circumferential and pitchwise locations for these pressure taps are shown in Table 3.

TABLE 3 Diffuser leading edge static pressure tap locations Pitchwise Circumferential Position Passage Description Position (deg) (%) No. P1 52.0 60 4 P2 85.1 90 6 P3 103.9 20 8 P4 165.8 50 12 P5 198.9 80 14 P6 217.6 10 16 P7 279.5 40 20 P8 312.7 70 22 P9 350.1 30 25

FIG. 9 shows the performance map of the compressor stage in terms of normalized total pressure ratio from 60% to 100% corrected speed from choke to high loading conditions. The operating conditions near the design loading conditions are indicated by the green circles. In the present study, the static pressure field at the diffuser leading edge was constructed near the nominal loading at design speed, indicated by a green circle at 100% corrected speed on the compressor map.

FIGS. 10a and 10b show the variation of the diffuser leading edge static pressure. In FIG. 10a , the static pressure is shown in terms of the absolute circumferential locations of the measurements. There is an approximately 40% peak-to-peak variation in the static pressure at the diffuser leading edge. There is no apparent trend in the measurements as shown in FIG. 10a . In contrast, the static pressure data is more informative when shown in terms of its pitchwise position, FIG. 10b . There is a higher static pressure close to the diffuser vane and low pressure between vanes, which agrees with previous findings in the prior art.

In the actual reduction to practice according to the present disclosure, a total of ten wavenumbers of interest were selected. These include the first two harmonics from the wakes at station 1 caused by the struts and rakes (Wn=4 and 8), the first five harmonics of the diffuser counts (W_(n)=25, 50, 75, 100, and 125), and the interactions between the compressor inlet struts and the vaned diffuser (W_(n)=21, 17, and 34). The condition numbers of the probe set for the 10 selected wavenumbers are shown in FIG. 11. The values of all the condition numbers fall in the range between 1.0 to 2.0 indicating the probe set is able to characterize all wavenumbers of interest. However, it is worth noting that this is a unique case. For instance, out of the multiple probe sets instrumented along the flow path at different stations (impeller exit, diffuser leading edge, etc.), only the probe set located at the diffuser leading edge yields a reasonable condition number.

Table 4 lists the values of Pearson's r, the fitting residual, as well as the rank of individual wavenumber. The wavenumber of 25 yields the best fitting results with highest value in Pearson's r and the lowest fitting residual. A single-wavelet approximation using Wn=25 yields a value for Pearson's r greater than 0.95. This indicates that the potential field of the vaned diffuser is dominant in the static pressure field at the diffuser leading edge. Furthermore, the wavenumber combinations yielding the best fitting results using single-, double-, and triple-wavelet approximations are listed in Table 5. As discussed previously, the static pressure field at the diffuser leading edge is primarily dominated by the potential of the diffuser vanes and, thus, a wavenumber of 25 yields the best fitting results for single-wavelet approximation. In addition, results indicate that the addition of the 2^(nd) harmonic wavenumber from the diffuser potential (Wn=50) yields the best fitting results for double-wavelet approximation. This agrees with the findings from Sanders and Fleeter showing that the variation in the static pressure field near the diffuser leading edge is dominated by the first few harmonics of the diffuser potential field. Finally, for the triple-wavelet approximation, the optimal wavenumber combination yielding the best fitting results is realized when including the effects from inlet strut-diffuser interactions (Wn=17), and the primary and 2^(nd) harmonic of the wavenumber from the diffuser potential field (Wn=25, 50).

TABLE 4 Rank of data fitting for a variety of wavenumber combinations Pearson Fitting Wave No. Correlation Residual Rank 4 0.302 8.711 4 8 0.265 8.833 6 17 0.416 8.311 3 21 0.469 8.068 2 25 0.973 2.105 1 34 0.177 8.992 10 50 0.250 8.848 8 75 0.297 8.725 5 100 0.244 8.860 9 125 0.259 8.825 7

TABLE 5 Best combination of wavelets from single-, double-, and triple-wavelet approximation Wave No. Pearson Fitting Combination Correlation Residual, % [25] 0.9731 2.2 [25, 50] 0.9927 1.1 [17, 25, 50] 0.9996 0.25

In addition, the deviation between the measurements and fitting results at individual sensor locations are shown in FIG. 12. There is a significant improvement in data fitting from the single-wavelet to triple-wavelet approximation. For instance, there is still an approximately 4% peak-to-peak deviation between the measurements and fitting data for the base case using the single-wavelet approximation method. The deviation drops to less than 2.5% by using the double-wavelet approximation method. Finally, the reconstructed signal using wavenumbers of [17, 25, and, 50] yields the best agreement with the measurements at all the sensor locations. The peak-to-peak deviation between the measurements and fitted data is less than 0.4%. The confidence of the fit in terms of Pearson's r is 99.7%, indicating no need to include extra wavelets.

Furthermore, the reconstructed pressure field at the leading edge from the triple-wavelet approximation method is shown in FIG. 13a . The measurements at all locations are also shown, indicated by the circles, and the circumferential coverage of individual diffuser passage at the leading edge is indicated by the shaded areas. The pressure field at the leading edge of the diffuser is determined by the constructive and destructive interactions between the three wavelets, as shown in FIG. 13b , and are the primary sources for the passage-to-passage variations. In the present case, the variation in the pressure field at the diffuser leading edge is dominated by the potential field from the diffuser vanes. As shown in FIG. 13c , the pressure variation due to the presence of diffuser vanes (Wn=25) is approximately 14% with respect to the pitch-averaged value. The second contribution to the static pressure variation is the 2^(nd) harmonic of the diffuser potential (Wn=50), yielding an approximate 4% variation in the circumferential direction at the diffuser leading edge. Finally, there is a small influence from the inlet strut-diffuser interactions (Wn=17), causing approximately 2% of variation in the circumferential pressure field at the diffuser leading edge.

It is worth noting that there is great value in understanding the content and interactions between each component in the pressure field upstream of the diffuser leading edge. For instance, in centrifugal compressors, one of the primary causes for impeller blade failure is the effect of the potential field from the vaned diffuser. In the present case, this corresponds to wavelets with wavenumbers of 25 and 50. The potential field imposes an unsteady pressure to the impeller blades as it passes by every diffuser passage. The magnitude of the wavelet determines the magnitude of the unsteady force acting on the impeller blades, which determines the vibration amplitude of the blade if the passing frequency is close to the blade natural frequency. Although the static pressure field upstream of the diffuser leading edge in a centrifugal compressor is used to illustrate the potential of the methods in addressing some of the forced response challenges, the method can also be applied to other types of turbomachines including axial compressors and radial and axial turbines.

In addition to reconstructing the detailed flow features, the method can also be used to obtain reliable mean flow properties for characterization of engine, component, or stage efficiency. Historically, the mean flow properties have been calculated using certain averaging methods. A variety of averaging methods have emerged during the past few decades including area-average, mass-average, work-average, and momentum-average methods. However, without the detailed information of flow properties around the full annulus, the accuracy of the averaged value as a representation of the true mean flow property is limited. Additionally, one challenge occurring in almost every engine test campaign is sensor mortality. In many cases, the measurements are not recoverable and, thus, result in increased instrumentation error and even larger uncertainty in the follow-on performance evaluation. Therefore, a robust method for probe arrangement and mean value calculation is of great value.

Table 6 lists the non-dimensional mean value of the diffuser leading edge static pressure obtained from circumferential-averaging, pitchwise-averaging, and triple-wavelet approximation methods. In the present case, all mean values obtained from the three methods are very close to each other. There is less than a 0.5% difference between the values from pitchwise-averaging and triple-wavelet approximation methods, and there is approximately a 3% difference between the values obtained from the circumferential-averaging and the triple-wavelet approximation methods.

TABLE 6 Comparison of mean diffuser le static pressure from different methods Pitchwise Circumferential Multi-wavelet Average Average Fitting 1.0 0.9695 1.0044

However, for cases with sensor dropout, the method using triple-wavelet approximation yields more repeatable values in comparison with the two averaging methods. For instance, as shown in FIG. 7a , for the case with one malfunctioning pressure measurement, the variation in the mean value of static pressure obtained using the triple-wavelet approximation method is less than 0.5%. In contrast, the variation in the mean static pressure obtained from the pitchwise-averaging method is 1.5 times larger, increasing to 2%. Additionally, the circumferential-averaging method yields the largest variation in calculation of the mean static pressure. There is an approximate 4.1% variation in the mean static pressure for cases with one sensor dropout. Furthermore, for cases with two sensor dropouts, the triple-wave approximation method still yields very repeatable mean values for the diffuser leading edge static pressure, with less than 1.2% in variation. In contrast, both averaging methods result in significantly larger variations in the mean values of the static pressure at the diffuser leading edge. For instance, the pitchwise-averaging method yields 3.7% variation while the circumferential-averaging method yields 8.1% variation in the mean value of the diffuser leading edge static pressure

In addition, the influence of this increased uncertainty (due to missing measurements) on the calculation of the static pressure recovery coefficient are investigated. The results are shown in FIG. 7b . In both cases, with one or two malfunctioning measurements, the method using multi-wavelet approximation method yields much smaller uncertainty in the calculated diffuser static pressure rise coefficient. The uncertainties in the diffuser static pressure coefficient caused by one or two missing measurements are 0.35% and 1.0%, respectively, using the mean value of static pressure from the triple-wavelet approximation method. In contrast, the uncertainties in the diffuser static pressure coefficient caused by one or two missing measurements are 1.7% and 3.1%, respectively, using the mean value of static pressure from the pitchwise-averaging method, and 3.2% and 6.3%, respectively, from the circumferential-averaging method.

Referring to FIGS. 14a and 14b , provided are comparisons of errors in calculating mean diffuser leading edge (FIG. 14a ) and static pressure and static pressure recovery coefficient (FIG. 14b ) using averaging methods and the multi-wavelet approximation method.

In the second experiment, a PAX100 compressor was used with a reduced vane count for stator 1 (denoted PAX101). The PAX100 compressor design features an IGV followed by three stages, shown in FIG. 15. All three of the rotors are integrally bladed and each stator row is uniquely manufactured as a 180-degree segment featuring a shroud on both the inner and outer diameters. Between each stage, instrumentation ports in the casing endwall allow for various probes to be inserted into the flow field. While the compressor is in operation, each stator row also has the capability to individually circumferentially traverse an angular distance up to approximately 15 degrees, or more than two stator vane passages. This enables vane clocking, allowing pitchwise measurements including wake traverses. The circumferential vane position is measured with precision string potentiometers.

In the original PAX100 configuration, the IGV, stator 1 (S1), and stator 2 (S2) all have the same vane count of 44 providing a unique environment to study the effects of vane clocking on compressor aerodynamic performance as well as rotor forced response. However, the effects of S1 and S2 on rotor 2 (R2) forced response are indistinguishable using this configuration. To address this challenge, a reduced-count Si vane row was designed with 38 vanes (19 vanes per 180-degree segment). However, with the introduction of a different, reduced-vane count for Si into the standard compressor configuration, the full annulus could not be captured in a single vane pass traverse. For the PAX100 configuration, with a uniform blade count of 44 for the IGV, S1, and S2, one vane passage could effectively characterize the entire annulus of the compressor (neglecting the effects of S3). For the PAX101 configuration, this characterization gets more complicated as one vane passage is no longer indicative of the entire annulus. To illustrate this complexity, a simple model was developed in the prior art to demonstrate how the blades would line up relative to one another around the annulus, FIG. 16. The circumferential positions for all blades from S1, S2/IGV (shown as a single row because of the same vane counts), and S3 are represented in red, blue, and green, respectively. In the PAX101 configuration, as all of the stator blade counts have a greatest common factor of 2, blades of all 4 stationary rows (IGV, S1, S2, and S3) only exactly line up every 180° (0° and 180° in FIG. 9). In addition, with decreasing the vane count of S1 by 6, the blades of IGV, S1, and S2 approximately line up every 60° (0°, 60°, 120°, 180°, 240°, and 300° in FIG. 9). Thus, a 6/rev pattern and a 2/rev pattern manifest around the annulus.

Since the instrumentation is stationary, S1 and S3 were clocked with respect to IGV and S2 in a particular fashion so as to imitate the location of a probe if it was able to be traversed around the annulus. These established 7 clocking configurations, labeled in FIG. 16, to map out a 60° segment of the annulus. The offset angles of S1 and S3 with respect to S2/IGV at all 7 clocking configurations are shown in FIG. 17. At each configuration, the position of IGV and S2 remains the same, while the position of S1 and S3 were adjusted with respect to IGV/S2 to match the relative blade spacing at the different annulus locations. Since the stator rows can only be traversed a finite difference, to achieve a full 60° sector, the clocking configuration orientation would have to be reversed just past the point when the difference between vanes exceeds 4.5° (configurations 5, 6, and 7 in FIG. 17).

With the clocking configurations in place, a comprehensive experimental campaign was conducted at 86% corrected speed on the 100% corrected speed peak efficiency loading line, shown in FIG. 18. This part-speed operation was selected to characterize the forcing functions for R2 forced response near the resonant crossing of the 38EO excitation of the R2 1^(st) torsion vibratory mode. In the experiment, seven-element total pressure rakes were placed behind S1, R2, and S2 at three different circumferential locations (noted as location A, B, and D). At each clocking configuration, all stators were traversed together over the length of a S1 vane passage at a resolution of 5% S1 passage, as indicated by the grayed area in FIG. 17. At the end of the test campaign, a total of 58.6° of effective travel around the annulus for each rake was achieved, and a total of 47.6% of the entire annulus was mapped using the three circumferentially placed rakes due to some overlap.

FIG. 19 shows the total pressure measurements acquired at mid-span downstream of S2. The measurements from rakes at location A, B, and D are represented by the blue, green, and brown lines. The measurements from traverses one to seven are also indicated in the figure. As expected, the stator-stator interactions result in complicated patterns of passage total pressure profile. There are evident passage-passage variations with a 6/rev feature, indicated by the dash line.

In the present disclosure, the total pressure downstream of Stator 2 is selected for flow reconstruction. Stator 2 is an embedded stage and, thus, provides an ideal environment to examine the potential of the method in characterizing the flow features associated with blade row interaction. The total pressure field downstream of S2 is dominated by the wakes from upstream stator rows, the potential field of downstream S3, and the interactions between these stator rows. According to the empirical guidelines provided previous section, a set of 20 wavenumbers were selected to reconstruct the total pressure field using the full dataset from experiments. The selected wavenumbers can be categorized into four types:

-   1. The first eight harmonics of IGV and S2 wakes     (Wn=[44,88,132,176,220,264,308,352]); -   2. The first four harmonics associated with reduced vane count of S1     (Wn=[38,76,114,152]); -   3. The first four harmonics of S3 potential field     (Wn=[50,100,150,200]); and -   4. The wavelets associated with IGV-S1-S2-S3 interactions     (Wn=[2,6,12,24]).

This set of wavelets yields a condition number of 3.15 using the full dataset from the experiment. The reconstructed total pressure profile (black) at mid-span downstream of S2 is shown in FIG. 20a . In addition, the total pressure profile acquired from experiments is plotted on top of the reconstructed total pressure profile for comparison. The reconstructed total pressure profile agrees well with the results from experiment. The features associated with passage-passage variations are resolved in the reconstructed total pressure profile. For instance, the 6/rev features due to the S1-S2/IGV interaction are characterized in the reconstructed profile. Despite some deviations in the depth of the pressure deficit between the reconstructed and measured total pressure profiles, there is a very good overall agreement achieved between the reconstructed and measured total pressure profiles.

Furthermore, efforts were made to reconstruct the total pressure profile using a reduced wavelet set. The considerations for selection of the reduced wavelet set are two-fold and need to be balanced. While a smaller number of wavelets would require fewer data points to reconstruct the flow, the reduced wavelet set should still be able to characterize the flow features of interest. After comparing the magnitudes of all the wavelets in the reconstructed total pressure profile, the first 12 dominant wavelets were selected: Wn=[6,12,38,50,44,88,132,176,220,264,308,352]. The reduced wavelet set includes all eight wavelets associated with S2/IGV wakes but eliminates the higher harmonics associated with S1, S3, and the stator-stator interactions. This reduced wavelet set yields a better condition number (1.69) using the full dataset from the experiment, as shown in Table 7. The total pressure at mid-span downstream of S2 was reconstructed using the first 12 dominant wavelets, and the results are shown in FIG. 20b . There are very small differences between the reconstructed total pressure profile using 20 wavelets and the reconstructed total pressure profile using 12 wavelets. Good agreement between the reconstructed total pressure profile and the results from experiment are achieved by using a reduced number of wavelets. Therefore, 12 wavelets are sufficient to reconstruct this total pressure profile at high fidelity.

TABLE 7 Condition number for the full dataset and reduced dataset Wave No. No. of Full Reduced Combination Wavelets Dataset Dataset [44 88 132 20 3.15 11.87 176 220 264 308 352 38 76 114 152 50 100 150 200 2 6 12 24] [6 12 38 50 12 1.69  1.97 44 88 132 176 220 264 308 352]

Additionally, an effort was made to reconstruct the total pressure field using a reduced data size, for instance, using data from three or four traverses instead of all seven traverses. To assure a high-fidelity result, an intelligent selection of the optimal traverse combinations included in the reduced dataset was exercised to achieve a small condition number. The selected reduced dataset contains three traverses from each rake including traverse numbers two, three, and six for the rake at location A; traverse numbers two, five, and seven for the rake at location B; and traverse numbers two, four, and seven for the rake at location D. This yields a condition number of 1.97 for the case when trying to resolve 12 wavelets. The reduced dataset accounts for 43% of the full dataset and only 20% of the full annulus coverage. The reconstructed total pressure field using the reduced dataset is shown in FIG. 20c . The segments of data used for flow reconstruction are indicated by the blue bands on the abscissa. Strong agreement in the total pressure profile between the reconstructed and experimental results is achieved in the passages where the experimental data are used for flow reconstruction. More importantly, good agreement is also achieved in the passages where the experimental data are not used for flow reconstruction. The features associated with passage-passage variations are nicely resolved in the reconstructed total pressure profile using the reduced dataset. There are very small differences between the two reconstructed total pressure profiles using the full and selected reduced datasets.

After exploring the influences of the number of wavelets and the size of the dataset, one important conclusion can be drawn: the full annulus total pressure profile downstream of S2 can be reconstructed with high-fidelity by using a small segment of the dataset and inclusion of a limited number of wavelets. Based on this finding, the total pressure profile at the near hub (12%) and near shroud (88%) are reconstructed using the reduced dataset with 12 wavelets. The results are shown in FIGS. 21a, 21b, and 21c . The reconstructed total pressure profile at 65% span is not shown in the figure due to the physical differences in the spanwise distribution of the pressure elements for each of the three rakes used (i.e. Rake B has a pressure element at 65% spanwise location while Rakes A and D have pressure elements at 70% instead). At the other six spanwise locations, the reconstructed total pressure profiles agree well with the results from experiment including the passages where the experimental data are not used for flow reconstruction. The patterns associated with passage-passage variations are nicely resolved in the reconstructed total pressure profile. Additionally, comparing to the midspan, there is better agreement in the depth of the pressure deficit associated with wakes between the reconstructed and measured total pressure profiles achieved at the near hub and near shroud regions using either the full or reduced dataset with inclusion of 12 wavelets. At all the spanwise locations, the mean total pressures obtained using the multi-wavelet method are almost identical to the values calculated using the pitchwise-averaging method, with the maximum deviation less than 0.025%. In addition, the multi-wavelet method yields very repeatable values for the mean total pressure for all cases (including the full or reduced datasets with inclusion of 20 or 12 wavelets).

TABLE 8 List of normalized mean total pressure at all spanwise locations Multi-wavelet Max Span, Approx. Dev, % Exp. F-20 F-10 R-10 % 12 1.1464 1.1464 1.1464 1.1463 0.0052 20 1.1473 1.1473 1.1474 1.1473 0.0071 35 1.1480 1.1481 1.1482 1.1482 0.0184 50 1.1468 1.1470 1.1471 1.1471 0.0238 70 1.1457 1.1457 1.1459 1.1458 0.0226 80 1.1437 1.1436 1.1439 1.1437 0.0231 88 1.1407 1.1407 1.1409 1.1407 0.0212

Referring to FIG. 22, a flowchart 2200 is provided to show overall steps of the method of reconstructing nonuniform circumferential flow in a turbomachine. As input to the method, wavenumber of interest, probe position, probe measurements, and wavelets of interest are initially provided to a processor, as indicated by block 2202. Thereafter, a multi-wavelet approximation is carried out (see equations (3), (4), and (5)), as indicated by block 2204. Thereafter, magnitude (A_(i)) and phase information (φ_(i)) for the wavelet of interest is determined, as indicated by block 2206, from which a reconstructed flow field is generated, as indicated by block 2208. Thereafter, a confidence measure is evaluated to determine confidence in the generated data, as indicated by block 2210.

Determining wavenumber: Even though the circumferential flow in compressors can be approximated using a few dominant wavelets, resolving all of these wavenumbers can still be challenging. In practice, due to the cost and blockage associated with each probe, there is usually a limit on the number of probes allowed per blade row. Typically, a range of 3 to 8 rakes/probes per blade row is achievable. However, according to Eqn. (3), a set of 4, 6, and 8 probes can resolve 1, 2, and 3 wavenumbers, respectively. Thus, an intelligent selection of the most important wavenumbers is needed to assure the best results for reconstructing the signal from a limited number of probes. The most important wavenumbers can be determined with the help of information from either reduced-order modeling or high-fidelity computational fluid dynamics simulations. For cases with no information available except for airfoil counts, recommended guidelines based on previous research of multi-stage interactions for representative wavenumber selection are:

-   -   1. Upstream and downstream vane counts;     -   2. Differences of the upstream and downstream vane counts;     -   3. Wavenumbers associated with low-count stationary component         (i.e. upstream and downstream struts for the front and rear         stages).         For instance, FIG. 23 is a schematic of a typical multi-stage         axial compressor used in high-pres sure compressor (HPC)         assembly of gas turbines. The dominant wavenumbers for the flow         downstream of stator 2 include: 1) 38 (stator 1 vane count), 44         (stator 2 vane count), and 50 (stator 3 vane count); 2) 8 (the         difference of upstream vane count) and 6 (the difference of the         downstream vane count); and 3) 4 (the vane count of the upstream         struts), which is not shown in the figure. Therefore, a         selection of 6 wavelets is appropriate. The process of selecting         wavelet and wavenumber is further demonstrated in Table 1,         below.

TABLE 1 Selection of wavelet and wavenumber No. of Wavelet Wavenumber Consideration Criterion 1 38 Upstream blade row 1st (S1) vane count 2 44 Vane count of itself 1st (S2) 3 50 Downstream blade row 1st (S3) vane count 4 8 S2 − S1 = 44 − 38 = 8 2nd 5 6 S3 − S2 = 50 − 44 = 6 2nd 6 4 Vane count of inlet 3rd struts As a result, 6 dominant wavelets can be selected along with wavenumbers are 38, 44, 50, 8, 6, 4, respectively.

Those having ordinary skill in the art will recognize that numerous modifications can be made to the specific implementations described above. The implementations should not be limited to the particular limitations described. Other implementations may be possible. 

1. A method for reconstructing nonuniform circumferential flow in a turbomachine, comprising: receiving one or more wavenumbers of interest; receiving positional information for a plurality of circumferential positions of a plurality of instrumentation probes; receiving signals from the plurality of instrumentation probes to generate a spatially under-sampled data; and from the spatially under-sampled data determining a multi-wavelet approximation reconstructing circumferential flow field.
 2. The method of claim 1, further comprising: evaluating confidence in the reconstructed circumferential flow field.
 3. The method of claim 2, wherein the evaluation of confidence is based on calculating a Pearson's correlation coefficient which represents a measure of linear correlation between actual measurements from the plurality of instrumentation probes and a fitted data from the multi-wavelet approximation at the plurality of circumferential positions.
 4. The method of claim 3, wherein the Pearson correlation coefficient is: $\rho = \frac{{\sum\limits_{j = 1}^{m}\;{x_{j}x_{{fit},j}}} - {\left( {\sum\limits_{j = 1}^{m}\;{x_{j}{\sum\limits_{j = 1}^{m}\; x_{{fit},j}}}} \right)\text{/}m}}{\sqrt{\left( {{\sum\limits_{j = 1}^{m}\; x_{j}^{2}} - {\left( {\sum\limits_{j = 1}^{m}\; x_{j}} \right)^{2}\text{/}m}} \right)}\left( {{\sum\limits_{j = 1}^{m}\; x_{{fit},j}^{2}} - {\left( {\sum\limits_{j = 1}^{m}\; x_{{fit},j}} \right)^{2}\text{/}m}} \right)}$ wherein x_(fit)(θ) represents predicted flow properties from the reconstructed signal, x(θ) represents the actual measurements; m represents the number of measurements, x_(j) is the measurement at the j^(th) circumferential location, and x_(fit,j) corresponds to the reconstructed value at the j^(th) circumferential location.
 5. The method of claim 3, wherein the Pearson correlation coefficient is near 0 where the linear correlation is highly uncorrelated and near 1 where the linear correlation is highly correlated.
 6. The method of claim 2, wherein the evaluation of confidence is based on calculating the multi-wavelet approximation at the plurality of circumferential positions.
 7. The method of claim 6, the confidence is based on root-mean-square of between actual measurements from the plurality of instrumentation probes and fitted data from the multi-wavelet approximation at the plurality of circumferential positions.
 8. The method of claim 7, where R_(rms) is: $R_{rms} = \sqrt{\frac{1}{m}\left( {\sum\limits_{j = 1}^{m}\;\left( {x_{{fit},j} - x_{j}} \right)^{2}} \right)}$ wherein x_(fit)(θ) represents predicted flow properties from the reconstructed signal, x(θ) represents the actual measurements; m represents the number of measurements, x_(j) is the measurement at the j^(th) circumferential location, and x_(fit,j) corresponds to the reconstructed value at the j^(th) circumferential location.
 9. The method of claim 1, wherein circumferential flow field is approximated as: x(θ)≈c ₀+Σ_(j=1) ^(N)(A _(i) sin(W _(n,i)θ+φ_(i))), where x(θ) represents flow property along the circumferential direction, c₀ represents the mean flow property of the nonuniform circumferential flow, W_(n,i) represents the i^(th) wavenumber, and A_(i) and φ_(i) represent the magnitude and phase of a wavelet of the i^(th) wavenumber. 